1,5

An Orloj clock sequence is a finite sequence of positive integers that, when iterated, can be grouped so that the groups sum to successive natural numbers. There is one unrefined sequence whose values sum to each n; all other Orloj clock sequences summing to n can be obtained by refining this one. Refining means splitting one or more terms into values summing to that term. (The unrefined sequence for n = 2^k*(2m-1) is the sequence for 2m-1 repeated 2^k times, but any single refinement - possible unless m = 1 - will produce an aperiodic sequence summing to n.) The Orloj clock sequence is the one summing to 15: 1,2,3,4,3,2, with a beautiful up and down pattern.

Let b(i),0<=i<k be all the residues of triangular numbers modulo n in order, with b(k)=n. The differences b(i+1)-b(i) are the sequence for n. The sequence for 2n is the sequence for n repeated.

For a sum of 5, we have 1,2,2, which groups as 1, 2, 2+1, 2+2, 1+2+2, 1+2+2+1, .... This could be refined by splitting the second 2, to give the sequence 1,2,1,1; note that when this is grouped, the two 1's from the refinement always wind up in the same sum.

The array starts: 1; 1,1; 1,2; 1,1,1,1; 1,2,2; 1,2,1,2; 1,2,3,1.

(PARI) {Orloj(n) = local(found, tri, i, last, r); found = vector(n, i, 0); found[n] = 1; tri = 0; for(i = 1, if(n%2==0, n-1, n\2), tri += i; if(tri >= n, tri -= n); found[tri] = 1); last = 0; r = []; for(i = 1, n, if(found[i], r = concat(r, [i-last]); last = i)); r}

Cf. A028355, A118382. Length of row n is A117484(n).

Sequence in context: A082478 A083382 A025900 this_sequence A115766 A108339 A138559

Adjacent sequences: A118380 A118381 A118382 this_sequence A118384 A118385 A118386

nonn,tabf

Frank Adams-Watters (FrankTAW(AT)Netscape.net), Apr 26 2006